TranscriptSolving Absolute Value Equations

Jasmine is on a plane, flying to her vacation destination. She's super excited about the trip! She bought a vacation package for an amazing price, but there may be one teeney weeney little problem with her plan… she has no idea where the plane will land. The vacation destination is a surprise.

All Jasmine knows is that there will be a 30° Fahrenheit temperature difference between her home and the vacation spot. At home, it's 60° Fahrenheit, so Jasmine imagines that soon she’ll be hanging out at a beach or exploring a jungle for lions and tigers. The airplane just landed, but wait...

Absolute value equations

Something’s not right…Looking out the window, Jasmine sees there’s no sun at all, only grey skies and lots of snow. She can’t figure out how she made such a mistake. Let's explore solving absolute value equations and figure out where Jasmine went wrong. First, we'll summarize what we know.

Summary of prior knowledge

The temperature at Jasmine's home is 60° Fahrenheit. We don’t know the temperature of the unknown vacation destination, so we'll use the variable 'x'. Jasmine is guaranteed that the temperature difference between her home and the vacation spot is 30° Fahrenheit. What she doesn’t realize is that the temperature difference could be 30° higher or 30° lower than at home.

First Example

She forgets that the absolute value of a number is always positive. Let's show her how to set up the equation to model this situation. The |x - 60| = 30. The expression inside the absolute value bars can equal 30 or -30, so to solve, we need to set up two different equations.
x - 60 = 30... ...and x - 60 = -30.

Now solve each equation by adding 60 to both sides of both equations, thereby isolating the variable 'x'. We're left with two possible solutions: x = 90°F......or x = 30° F. Remember, it’s always a good idea to check your work. (pause...) Poor Jasmine, she planned for a vacation in 90° weather but arrived at a destination with 30° weather! No wonder she's freezing rather than frolicking in the sun.

Second Example

Now that you get the concept, let’s solve another absolute value equation. The |4x + 20| = 100. Remember to set up two different equations, one equaling a positive value, and the other equaling a negative value. 4x + 20 = 100 and 4x + 20 = -100.. Subtract 20 from both sides of each equation, isolating the term 4x....then divide by 4 on both sides of each equation.

Positive and negative equations

For the positive equation, x = 20 and for the negative equation, x = -30. It's always a good idea to check your work to make sure both solutions work. Make sure you plug the right solution into the right equation. For the positive equation, plug in 20 for x. And for the negative equation, plug in -30 for x.

Solutions

Both solutions work, so you're good to go. The two possilble solutions for this absolute value equation are 20 or -30. You know practice makes perfect, so let’s do one more problem. The |x - 2| + 8 = 2. To solve, first isolate the absolute value before setting up the two different equations. Subtract 8 from both sides of the equation, so now the |x - 2| = -6. Negative six?

But wait, absolute value is always a positive number, never negative, so for this problem, there's no possible solution. Poor Jasmine. Standing in the snow, she’s simply shivering in her sandals. Oh look…Seems like the temperature might be warming up for Jasmine.

About this Video Lesson

Description

Just to refresh your memory, absolute value is the distance of a number from zero. We can also think of it as the magnitude of a number without considering the sign. The absolute value of a number is always a positive number, and the absolute value of zero is zero. You can add, subtract, multiply, and divide with the absolute value of a number. But, how do you solve for the variable of an equation when the variable is part of an expression inside absolute value bars?

The absolute value of the variable expression must be a positive number, but the value of the variable expression inside the bars can be negative or positive. So, if you are trying to determine the value of the variable inside the bars, you must set up two different equations, one positive and the other negative. The answer is one or the other, so the solution can have two different answers.

You will want to verify that both answers work by substituting them back into the corresponding equation. Be careful not to mix up the equations. We use absolute value all the time in the real world. Think about this statement, “Wow, your weight really changed!” This statement can take on two different meanings. Does the statement mean you lost weight or gained weight? It could go either way, just like the value of a variable inside absolute value bars. For more on this, watch the video.